Inequalities and Their Solutions
Write, solve, and graph one- and two-step inequalities, and interpret solution sets in real-world contexts.
Learning Objectives
- 1Translate real-world situations into one- and two-step inequalities
- 2Solve one- and two-step inequalities using inverse operations, including flipping the inequality sign when multiplying or dividing by a negative
- 3Graph solution sets on a number line using open and closed circles
- 4Interpret and communicate the meaning of an inequality's solution in context
More Than One Answer
An equation says two expressions are exactly equal. An inequality says one expression is larger than, smaller than, or at most/at least another — and instead of one answer, the solution is often an entire range of values.
You already solve inequalities informally every day: "I need at least $20 to see the movie and buy popcorn." That situation is not an equation — it has a whole set of correct amounts.
Inequality Symbols
| Symbol | Meaning | Graph |
|---|---|---|
> |
greater than | open circle, arrow right |
< |
less than | open circle, arrow left |
≥ |
greater than or equal to | closed circle, arrow right |
≤ |
less than or equal to | closed circle, arrow left |
Open circles mean the boundary value is not included. Closed circles mean it is included.
Example: x > 3 is graphed with an open circle at 3 and an arrow pointing right. The number 3 itself is not a solution; 3.001 is.
Example: x ≤ -1 is graphed with a closed circle at -1 and an arrow pointing left. The number -1 is a solution.
Think About
You need a grade of at least 90 to earn an A. Write this as an inequality using the variable g. Should the boundary be open or closed? Why?
One-Step Inequalities
Solving works exactly like equations — apply inverse operations to isolate the variable. There is one critical exception: multiplying or dividing by a negative number flips the inequality sign.
Example: x + 8 > 15
Subtract 8 from both sides: x > 7
Graph: open circle at 7, arrow right.
Example: -3x ≤ 12
Divide both sides by -3 and flip the sign: x ≥ -4
Graph: closed circle at -4, arrow right.
Why does the sign flip? Consider 6 > 4. Multiply both sides by -1: -6 and -4. On the number line, -6 is to the left of -4, so -6 < -4. The relationship reversed.
❓Concept Check
Solve and graph: -5x > 20. What is the solution set?
▸
Concept Check
Solve and graph: -5x > 20. What is the solution set?
Divide both sides by -5 and flip the sign: x < -4. Graph: open circle at -4, arrow pointing left.
Two-Step Inequalities
Same strategy as two-step equations: undo addition/subtraction first, then multiplication/division.
Example: 2x - 7 ≥ 3
Step 1: Add 7 to both sides. 2x ≥ 10.
Step 2: Divide by 2 (positive — no flip). x ≥ 5.
Graph: closed circle at 5, arrow right.
Example: -x/4 + 2 < 6
Step 1: Subtract 2. -x/4 < 4.
Step 2: Multiply both sides by -4 and flip the sign: x > -16.
Graph: open circle at -16, arrow right.
Worked example with check: A candidate needs more than 500 votes to win. She currently has 312. How many more does she need?
Let v = additional votes: 312 + v > 500
v > 188
She needs more than 188 additional votes. Check: 312 + 189 = 501 > 500. Correct.
Think About
Compare the process of solving 2x - 7 = 3 (an equation) to solving 2x - 7 ≥ 3 (an inequality). At what point do the processes diverge, and what is different about the answer you get?
Interpreting Solutions in Context
The solution to an inequality needs to make sense in the situation.
Example: A roller coaster requires riders to be at least 48 inches tall. Let h = height in inches.
h ≥ 48
But h also cannot be 0 or negative in this context. The solution in context is h ≥ 48 where h is a reasonable height for a person.
Example: A student can spend at most $35 on school supplies. Notebooks cost $3 each. How many notebooks n can she buy?
3n ≤ 35
n ≤ 11.67
Since you cannot buy a fraction of a notebook, the contextual answer is n ≤ 11 (n is a whole number).
❓Concept Check
A parking garage charges $4 per hour. You have $18. Write and solve an inequality for the number of hours h you can park. Interpret the answer in context.
▸
Concept Check
A parking garage charges $4 per hour. You have $18. Write and solve an inequality for the number of hours h you can park. Interpret the answer in context.
4h ≤ 18. Divide by 4: h ≤ 4.5. In context, since parking is billed by the hour, you can park for at most 4 full hours.
Practice: Khan Academy's "One-step inequalities" and "Two-step inequalities" exercises match exactly what you need here. Find them at 7th grade math (TX). Work through both sets, paying special attention to word problems.
Think About
An equation like 2x - 7 = 3 has exactly one solution. An inequality like 2x - 7 ≥ 3 has infinitely many solutions. Yet both are solved by the same algebraic steps. What does this tell you about the relationship between equations and inequalities?
Looking Ahead
Inequalities describe ranges rather than single values — and that idea will reappear throughout algebra. In Unit 7, when you study slope and linear relationships, you will graph lines on a coordinate plane. Eventually in Algebra I, you will extend this work to graphing linear inequalities on the plane, producing shaded half-planes. Every rule you practice here — especially the sign flip — carries directly forward.